A probabilistic rule of mixtures for elastic moduli

A novel approach is developed for mathematically modeling the variability o bserved in experimentally determined elastic moduli of longitudinally orien ted fibrous tissues such as ligaments and tendons. The elastic modulus of t hese tissues is modeled with a rule of mixtures (ROM) where each parameter (fibril and matrix moduli and fibril volume fraction) is assumed to be an i ndependent random variable. A joint density Function formed from the indepe ndent densities results in a probabilistic ROM (pROM). This pROM is used to generate a distribution of moduli which agrees well with moduli determined from tests of rabbit medial collateral ligaments (Woo and Ohland, 1994, Un published experimental data as gift). Minimizing the error between the pROM and experimental distributions resulted in an integrated error of 9% for a constrained set of independent distribution parameters derived from the li terature. This pROM thus incorporates microstructural observations (fibril and matrix moduli and fibril volume fraction) to partially explain the expe rimentally observed variability in a macroscopic property (tissue modulus). (C) 1999 Elsevier Science Ltd. All rights reserved.